continuity the man-in-the-street understanding of a continuous process is something that proceeds...
Post on 19-Dec-2015
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CONTINUITYThe man-in-the-street understanding of a
continuous process is something that proceeds
smoothly, without breaks or interruptions.
Consequently, for a function to be
called “continuous”, we would expect its graph to
be a smooth line, without breaks or interruptions.
Let’s look at some graphs we would definitely not call continuous ; the best way to define “day” is
to think of “night”, “happy” is meaningless unless
you know “sad”, most concepts are better understood via their opposites !
Here is a function whose graph you would definitely not call continuous, it jumps at every integer!
The formal definition of is
(The notation is somewhat different from the textbook’s, it means the greatest integer ≤ )Here is the graph
There’s a break at every integer! What’s the trouble? Here is another
A break at 3 again! Two more graphs.
A hole at 2 !
On the right the hole has been incorrectly filled.The next example is the messiest.
Talk about not smooth! A little better:
What do these pictures tell us about our intuitive notion of a continuous graph?
There should be:
No holes
No jumps
No uncertainties.
To a mathematician these mean:
(the order is mixed up.)
These three are condensed in:
And formally:
CONTINUITY AT
Definition. The function is said tobe continuous at if
(all three previous conditions are assured by this statement.)
Now by application of the three statements
No. 1 If , where and
are polynomials, and then
we get that
every rational function is continuous at every point where it is defined.
No. 2 (usual caveat about n) gives us that
radicals of continuous functions are continuous wherever the are defined.Finally, fromNo.3We get thatAll trigonometric functions are continuous wherever they are defined.Finally, if and are continuous atthen so are , , and if
What about the composition ?A look at this picture tells us that
If is continuous at and is continuous
at then is continuous at
Intermediate Value Theorem
Probably the most important (useful) property of continuous functions is the following
Theorem. If is continuous at every , then for every number
between and there is at least one such that
A picture will help:
Here is the situation:
You can’t join the two red dots “continuously without crossing the blue dotted line.